Arithmetic Progressions | Class 10 Mathematics Notes

5.3 nth Term of an AP

This section explains how to find the nth term of an Arithmetic Progression (AP) efficiently, using the example of Reena's salary increment introduced earlier. Reena starts with a salary of ₹8000 and receives an annual increment of ₹500.

Calculations for the 2nd, 3rd, 4th, and 5th year salaries are shown step-by-step by adding ₹500 repeatedly to the previous year's salary, leading to the expressions:

• 2nd year: 8000 + 500
• 3rd year: 8000 + 2 × 500
• 4th year: 8000 + 3 × 500
• 5th year: 8000 + 4 × 500

This pattern suggests that the salary for the nth year can be expressed as:

Salary for nth year = First salary + (n - 1) × Annual increment

Generalizing, for an AP with first term a and common difference d, the nth term an is given by:

an = a + (n - 1)d

The section provides several examples to illustrate this formula:

• Example 3: Find the 10th term of AP 2, 7, 12, ...
• Example 4: Find which term of AP 21, 18, 15, ... is -81 and whether any term is zero.
• Example 5: Determine the AP given the 3rd and 7th terms.
• Example 6: Check if 301 is a term of AP 5, 11, 17, 23, ...
• Example 7: Find how many two-digit numbers are divisible by 3.
• Example 8: Find the 11th term from the last term of AP 10, 7, 4, ..., -62.
• Example 9: Calculate simple interest each year and check if the interests form an AP.
• Example 10: Find the number of rows in a flower bed where the number of rose plants forms an AP.

The section also discusses alternative methods, such as reversing the AP to find terms from the end. It emphasizes the importance of understanding the formula for the nth term to quickly find any term without listing all preceding terms.

This formula is fundamental for solving various problems involving APs, including real-life applications like salary increments, production quantities, and financial calculations.

📊 Diagram: Fig. 5.1 illustrating ladder rungs is referenced for real-life AP examples.

🧪 Activity: Calculate nth terms for given APs and verify if certain numbers are terms of an AP.

🔗 Connection: Leads to the study of the sum of first n terms of an AP in section 5.4.